Volume 12, pp. 113-133, 2001.

Geršgorin-type eigenvalue inclusion theorems and their sharpness

Richard S. Varga

Abstract

Here, we investigate the relationships between $\mathcal{G}(A)$, the union of Geršgorin disks, $\mathcal{K}(A)$, the union of Brauer ovals of Cassini, and $\mathcal{B}(A)$, the union of Brualdi lemniscate sets, for eigenvalue inclusions of an $n\times n$ complex matrix $A$. If $\sigma (A)$ denotes the spectrum of $A$, we show here that $$\sigma (A)\subseteq \mathcal{B}(A)\subseteq \mathcal{K}(A)\subseteq \mathcal{G}(A)$$ is valid for any weakly irreducible $n\times n$ complex matrix $A$ with $n\ge 2$. Further, it is evident that $\mathcal{B}(A)$ can contain the spectra of related $n\times n$ matrices. We show here that the spectra of these related matrices can fill out $\mathcal{B}(A)$. Finally, if ${\mathcal{G}}^{\mathcal{R}}(A)$ denotes the minimal Geršgorin set for $A$, we show that $${\mathcal{G}}^{\mathcal{R}}(A)\subseteq \mathcal{B}(A).$$

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Key words

Geršgorin disks, Brauer ovals of Cassini, Brualdi lemniscate sets, minimal Geršgorin sets.

AMS subject classifications

15A18.

Links to the cited ETNA articles

[7]	Vol. 8 (1999), pp. 15-20 Richard S. Varga and Alan Krautstengl: On Geršgorin-type problems and ovals of Cassini